Mechanistic simulations of population dynamics

We simulated antagonistic population dynamics using a discrete Lotka-Volterra competition model, where each population’s growth rate is influenced by the fluctuations in a competitor population. Pairs of populations were simulated according to:

\[\begin{equation} N_i(t+1) = N_i(t) + \lambda_i * N_i(t) \end{equation}\]

where \(N_i\) is the size of population \(i\), \(t\) is each time step, and \(\lambda\) is the true growth rate of population \(i\). The true growth rate (\(\lambda\)) is determined by:

\[\begin{equation} \lambda_i = r*(1-\frac{(N_i(t) + \alpha_{ji}N_j(t))}{K_i}) \end{equation}\]

where \(r\) is population \(i\)’s maximum growth rate, \(\alpha_{ji}\) is the interaction effect of population \(j\) on population \(i\), \(N_j\) is the size of population \(j\), and \(K_i\) is the carrying capacity of population \(i\).

To test index sensitivity to error, we introduced process and observation error into the generated population time series.

Table 1: Constant simulation parameters
Value
Number of population pairs 10.0
Time steps 10.0
Initial size (N0) 100.0
Maximum growth rate (r) 1.5

Biodiversity change scenarios

Direction of change

To test index sensitivity to direction of change, we generated three scenarios of population-level biodiversity change.

We varied carrying capacity through time to drive populations to:

    1. decline in response to decreasing carrying capacity, representing situations where habitats are lost or altered,
    1. remain stable in response to constant carrying capacity, where habitats are left untouched, or
    1. increase in response to increasing carrying capacity, where habitat quality and/or availability increases through time.

In each scenario, carrying capacity begins at \(K = 100\) and each population’s initial size is exactly at this carrying capacity (\(N_0 = 100\)).

Carrying capacity trends used to generate the biodiversity change scenarios.

Figure 1: Carrying capacity trends used to generate the biodiversity change scenarios.

Covariance

To test whether the LPI is sensitive to lags in the covariance between populations, we then generated two levels of lag in the interaction effect dictating how populations covary through time:

  • The first level was lag-1, where the interaction effect of populations \(i\) on populations \(j\) was dependent on the abundance of populations \(i\) at the previous time step (\(t-1\)).
  • The second level was lag-2, where this interaction effect depended instead on the abundance of populations \(i\) from two time steps ago (\(t-2\)).

Measuring index sensitivity

For each biodiversity change scenario, index sensitivity was measured as the accuracy and precision of the calculated Living Planet Index when compared to the true biodiversity change trend. This true LPI trend was obtained by calculating the index values through time based on a population generated from the same parameters we used to generate populations for each scenario, without any added process error.

We measured accuracy as the difference between the calculated LPI and true LPI trend at each time step:

\[accuracy = LPI_{calculated} - LPI_{true}\]

As a first metric of precision, we evaluated whether the true LPI trend fell within the 95% confidence interval around the calculated LPI trend obtained from bootstrap resampling at each time step. As a second precision metric, we measured precision as the difference between the expected uncertainty interval and the width of the 95% confidence interval. The expected uncertainty interval is determined based on the process and observation errors introduced into the simulated population trends. I calculated each population’s growth rate with and without introduced error, and took the difference between the two to get a measure of the total introduced uncertainty in the growth rates. Then, I converted this uncertainty to the same scale as the LPI:

\[ precision = LPI_{CIwidth} - LPI_{uncertainty} \]

Is the LPI sensitive to direction of change?

Table 2: Simulation parameters
i j
Interaction effect (alpha) 0 0
Observation error +/- 5 +/- 5
Process error +/- 0, 0.1, 0.2 +/- 0, 0.1, 0.2
Lag 0 0

Process error = 0

True and calculated LPI trends for each direction of biodiversity change.

Figure 2: True and calculated LPI trends for each direction of biodiversity change.

Process error = 0.1

True and calculated LPI trends for each direction of biodiversity change.

Figure 3: True and calculated LPI trends for each direction of biodiversity change.

Process error = 0.2

True and calculated LPI trends for each direction of biodiversity change.

Figure 4: True and calculated LPI trends for each direction of biodiversity change.

Summary

Process error = 0

LPI sensitivity to direction of change.

Figure 5: LPI sensitivity to direction of change.

Process error = 0.1

LPI sensitivity to direction of change.

Figure 6: LPI sensitivity to direction of change.

Process error = 0.2

LPI sensitivity to direction of change.

Figure 7: LPI sensitivity to direction of change.

Summary table

direction Process_error mean_accuracy mean_precision
decline 0 -0.01 -0.213
decline 0.1 0.00 -0.222
decline 0.2 0.00 -0.221
stable 0 -0.01 -0.216
stable 0.1 0.00 -0.224
stable 0.2 0.00 -0.224
growth 0 -0.02 -0.221
growth 0.1 -0.01 -0.226
growth 0.2 -0.01 -0.232

Ability to capture the true LPI

Figure 8: Ability to capture the true LPI

direction Process_error true_within_rlpiCI n perc
decline 0 Success 11 1.0000000
decline 0.1 Success 11 1.0000000
decline 0.2 Success 11 1.0000000
stable 0 Success 10 0.9090909
stable 0.1 Success 11 1.0000000
stable 0.2 Success 11 1.0000000
growth 0 Success 4 0.3636364
growth 0.1 Success 4 0.3636364
growth 0.2 Success 9 0.8181818
Ability to capture the true LPI

Figure 9: Ability to capture the true LPI

Is the LPI sensitive to covariance?

To test whether the LPI is sensitive to covariance between populations, we generated biodiversity change scenarios with two types of interaction effects (\(\alpha\)) between populations:

  • For the first scenario, we set a negative interaction effect to represent an antagonistic interaction between populations \(i\) and \(j\), such as a competitive or predator-prey relationship, which causes negative covariance.
  • The second scenario was based on a positive interaction effect of the same magnitude(s) to replicate a positive interaction between populations \(i\) and \(j\), and therefore positive covariance.

Each type of interaction was tested at two interaction strengths. All interaction effects were immediate, meaning there was no time lag in the covariance in the populations’ growth rates.

Table 3: Simulation parameters
i j
Interaction effect (alpha) 0 0
Observation error +/- 5 +/- 5
Process error +/- 0, 0.1, 0.2 +/- 0, 0.1, 0.2
Lag 0 0

Process error = 0

True and calculated LPI trends for each scenario.

Figure 10: True and calculated LPI trends for each scenario.

Process error = 0.1

True and calculated LPI trends for each scenario.

Figure 11: True and calculated LPI trends for each scenario.

Process error = 0.2

True and calculated LPI trends for each scenario.

Figure 12: True and calculated LPI trends for each scenario.

Summary

Process error = 0

LPI sensitivity to direction of change and covariance.

Figure 13: LPI sensitivity to direction of change and covariance.

Process error = 0.1

LPI sensitivity to direction of change and covariance.

Figure 14: LPI sensitivity to direction of change and covariance.

Process error = 0.2

LPI sensitivity to direction of change and covariance.

Figure 15: LPI sensitivity to direction of change and covariance.

Summary table

direction interaction mean_accuracy se_accuracy mean_precision se_precision
decline Strong Asynchrony -0.004 0.001 -0.357 0.039
decline Weak Asynchrony -0.016 0.002 -0.257 0.030
decline No Synchrony -0.006 0.002 -0.213 0.026
decline Weak Synchrony -0.009 0.001 -0.208 0.025
decline Strong Synchrony -0.009 0.001 -0.206 0.025
stable Strong Asynchrony -0.018 0.003 -0.346 0.038
stable Weak Asynchrony -0.024 0.003 -0.250 0.029
stable No Synchrony -0.010 0.002 -0.216 0.026
stable Weak Synchrony -0.014 0.002 -0.206 0.025
stable Strong Synchrony -0.015 0.002 -0.205 0.025
growth Strong Asynchrony -0.030 0.004 -0.335 0.037
growth Weak Asynchrony -0.029 0.004 -0.259 0.030
growth No Synchrony -0.023 0.003 -0.221 0.027
growth Weak Synchrony -0.012 0.002 -0.215 0.026
growth Strong Synchrony -0.011 0.002 -0.215 0.026
Ability to capture the true LPI

Figure 16: Ability to capture the true LPI

direction interaction Process_error true_within_rlpiCI n perc
decline Strong Asynchrony 0 Success 11 1.0000000
decline Strong Asynchrony 0.1 Success 11 1.0000000
decline Strong Asynchrony 0.2 Success 11 1.0000000
decline Weak Asynchrony 0 Success 4 0.3636364
decline Weak Asynchrony 0.1 Success 11 1.0000000
decline Weak Asynchrony 0.2 Success 11 1.0000000
decline No Synchrony 0 Success 11 1.0000000
decline No Synchrony 0.1 Success 11 1.0000000
decline No Synchrony 0.2 Success 11 1.0000000
decline Weak Synchrony 0 Success 7 0.6363636
decline Weak Synchrony 0.1 Success 8 0.7272727
decline Weak Synchrony 0.2 Success 11 1.0000000
decline Strong Synchrony 0 Success 8 0.7272727
decline Strong Synchrony 0.1 Success 11 1.0000000
decline Strong Synchrony 0.2 Success 11 1.0000000
stable Strong Asynchrony 0 Success 4 0.3636364
stable Strong Asynchrony 0.1 Success 11 1.0000000
stable Strong Asynchrony 0.2 Success 11 1.0000000
stable Weak Asynchrony 0 Success 3 0.2727273
stable Weak Asynchrony 0.1 Success 11 1.0000000
stable Weak Asynchrony 0.2 Success 11 1.0000000
stable No Synchrony 0 Success 10 0.9090909
stable No Synchrony 0.1 Success 11 1.0000000
stable No Synchrony 0.2 Success 11 1.0000000
stable Weak Synchrony 0 Success 6 0.5454545
stable Weak Synchrony 0.1 Success 7 0.6363636
stable Weak Synchrony 0.2 Success 11 1.0000000
stable Strong Synchrony 0 Success 6 0.5454545
stable Strong Synchrony 0.1 Success 8 0.7272727
stable Strong Synchrony 0.2 Success 11 1.0000000
growth Strong Asynchrony 0 Success 4 0.3636364
growth Strong Asynchrony 0.1 Success 11 1.0000000
growth Strong Asynchrony 0.2 Success 11 1.0000000
growth Weak Asynchrony 0 Success 3 0.2727273
growth Weak Asynchrony 0.1 Success 9 0.8181818
growth Weak Asynchrony 0.2 Success 7 0.6363636
growth No Synchrony 0 Success 4 0.3636364
growth No Synchrony 0.1 Success 4 0.3636364
growth No Synchrony 0.2 Success 9 0.8181818
growth Weak Synchrony 0 Success 8 0.7272727
growth Weak Synchrony 0.1 Success 10 0.9090909
growth Weak Synchrony 0.2 Success 11 1.0000000
growth Strong Synchrony 0 Success 8 0.7272727
growth Strong Synchrony 0.1 Success 11 1.0000000
growth Strong Synchrony 0.2 Success 11 1.0000000 

The LPI is sensitive to covariance when trends are declining:  

* The LPI is particularly sensitive to negative covariance in declining trends, which are very overestimated (alpha = 0.1 and 0.2)  
* growing and stable trends are slightly overestimated when there is high positive covariance  
* however, this sensitivity does not change at different levels of process error

Is the LPI sensitive to lagged covariance?

We introduced two levels of lag in the interaction effect that influences how populations covary through time:

  • The first level was lag-1, where the interaction effect of populations \(i\) on populations \(j\) was dependent on the abundance of populations \(i\) at the previous time step (\(t-1\)).
  • The second level was lag-2, where this interaction effect depended instead on the abundance of populations \(i\) from two time steps ago (\(t-2\)).

Both of these lags were introduced into positively and negatively covarying populations shown above.

Table 4: Simulation parameters
i j
Interaction effect (alpha) 0 0
Observation error +/- 5 +/- 5
Process error +/- 0, 0.1, 0.2 +/- 0, 0.1, 0.2
Lag 0, 1, 2 0, 1, 2

Positive covariance

Process error = 0

Process error = 0.1

Process error = 0.2

Negative covariance

Process error = 0

Process error = 0.1

Process error = 0.2

Summary

Process error = 0

Process error = 0.1

Process error = 0.2

The LPI is not sensitive to lags of 1 or 2 time steps.

Summary table

direction interaction Lag Process_error mean_accuracy se_accuracy mean_precision se_precision
decline Strong Asynchrony 0 0 -0.004 0.001 -0.357 0.039
decline Strong Asynchrony 0 0.1 -0.006 0.001 -0.307 0.038
decline Strong Asynchrony 0 0.2 -0.008 0.001 -0.406 0.056
decline Strong Asynchrony 1 0 -0.002 0.000 -0.368 0.049
decline Strong Asynchrony 1 0.1 -0.004 0.001 -0.422 0.053
decline Strong Asynchrony 1 0.2 -0.007 0.001 -0.502 0.072
decline Strong Asynchrony 2 0 -0.002 0.000 -0.355 0.039
decline Strong Asynchrony 2 0.1 0.006 0.001 -0.392 0.047
decline Strong Asynchrony 2 0.2 0.006 0.002 -0.506 0.065
decline Weak Asynchrony 0 0 -0.016 0.002 -0.257 0.030
decline Weak Asynchrony 0 0.1 -0.005 0.001 -0.194 0.025
decline Weak Asynchrony 0 0.2 -0.003 0.001 -0.214 0.029
decline Weak Asynchrony 1 0 -0.007 0.001 -0.258 0.032
decline Weak Asynchrony 1 0.1 -0.005 0.001 -0.297 0.034
decline Weak Asynchrony 1 0.2 -0.007 0.001 -0.322 0.038
decline Weak Asynchrony 2 0 -0.009 0.001 -0.254 0.029
decline Weak Asynchrony 2 0.1 0.001 0.001 -0.257 0.031
decline Weak Asynchrony 2 0.2 0.001 0.000 -0.295 0.036
decline Weak Synchrony 0 0 -0.009 0.001 -0.208 0.025
decline Weak Synchrony 0 0.1 -0.005 0.001 -0.218 0.027
decline Weak Synchrony 0 0.2 -0.004 0.001 -0.241 0.031
decline Weak Synchrony 1 0 -0.007 0.001 -0.214 0.026
decline Weak Synchrony 1 0.1 -0.002 0.001 -0.221 0.028
decline Weak Synchrony 1 0.2 0.001 0.000 -0.259 0.030
decline Weak Synchrony 2 0 -0.008 0.001 -0.209 0.025
decline Weak Synchrony 2 0.1 -0.002 0.001 -0.184 0.021
decline Weak Synchrony 2 0.2 -0.003 0.001 -0.205 0.025
decline Strong Synchrony 0 0 -0.009 0.001 -0.206 0.025
decline Strong Synchrony 0 0.1 -0.003 0.000 -0.391 0.060
decline Strong Synchrony 0 0.2 -0.004 0.001 -0.611 0.093
decline Strong Synchrony 1 0 -0.004 0.001 -0.256 0.031
decline Strong Synchrony 1 0.1 -0.004 0.001 -0.419 0.058
decline Strong Synchrony 1 0.2 -0.016 0.004 -0.578 0.088
decline Strong Synchrony 2 0 -0.004 0.001 -0.244 0.029
decline Strong Synchrony 2 0.1 0.000 0.001 -0.346 0.051
decline Strong Synchrony 2 0.2 0.003 0.001 -0.536 0.085
stable Strong Asynchrony 0 0 -0.018 0.003 -0.346 0.038
stable Strong Asynchrony 0 0.1 -0.009 0.001 -0.288 0.035
stable Strong Asynchrony 0 0.2 -0.004 0.001 -0.376 0.051
stable Strong Asynchrony 1 0 -0.005 0.001 -0.365 0.045
stable Strong Asynchrony 1 0.1 -0.005 0.001 -0.429 0.051
stable Strong Asynchrony 1 0.2 -0.009 0.002 -0.509 0.068
stable Strong Asynchrony 2 0 -0.007 0.001 -0.355 0.038
stable Strong Asynchrony 2 0.1 0.005 0.001 -0.395 0.045
stable Strong Asynchrony 2 0.2 0.004 0.001 -0.500 0.059
stable Weak Asynchrony 0 0 -0.024 0.003 -0.250 0.029
stable Weak Asynchrony 0 0.1 -0.010 0.001 -0.193 0.025
stable Weak Asynchrony 0 0.2 -0.005 0.001 -0.209 0.028
stable Weak Asynchrony 1 0 -0.010 0.002 -0.255 0.031
stable Weak Asynchrony 1 0.1 -0.007 0.001 -0.300 0.034
stable Weak Asynchrony 1 0.2 -0.009 0.001 -0.325 0.038
stable Weak Asynchrony 2 0 -0.020 0.003 -0.252 0.029
stable Weak Asynchrony 2 0.1 -0.005 0.001 -0.260 0.031
stable Weak Asynchrony 2 0.2 -0.002 0.000 -0.299 0.035
stable Weak Synchrony 0 0 -0.014 0.002 -0.206 0.025
stable Weak Synchrony 0 0.1 -0.005 0.001 -0.222 0.027
stable Weak Synchrony 0 0.2 -0.003 0.001 -0.240 0.030
stable Weak Synchrony 1 0 -0.013 0.002 -0.209 0.026
stable Weak Synchrony 1 0.1 -0.005 0.001 -0.215 0.026
stable Weak Synchrony 1 0.2 -0.002 0.001 -0.238 0.028
stable Weak Synchrony 2 0 -0.014 0.002 -0.204 0.025
stable Weak Synchrony 2 0.1 -0.003 0.001 -0.184 0.021
stable Weak Synchrony 2 0.2 -0.003 0.001 -0.197 0.024
stable Strong Synchrony 0 0 -0.015 0.002 -0.205 0.025
stable Strong Synchrony 0 0.1 -0.005 0.001 -0.317 0.042
stable Strong Synchrony 0 0.2 -0.003 0.000 -0.469 0.058
stable Strong Synchrony 1 0 -0.008 0.002 -0.231 0.029
stable Strong Synchrony 1 0.1 0.000 0.000 -0.342 0.043
stable Strong Synchrony 1 0.2 -0.012 0.002 -0.476 0.061
stable Strong Synchrony 2 0 -0.011 0.002 -0.222 0.025
stable Strong Synchrony 2 0.1 0.001 0.001 -0.271 0.034
stable Strong Synchrony 2 0.2 0.004 0.001 -0.413 0.053
growth Strong Asynchrony 0 0 -0.030 0.004 -0.335 0.037
growth Strong Asynchrony 0 0.1 -0.020 0.002 -0.274 0.033
growth Strong Asynchrony 0 0.2 -0.022 0.002 -0.357 0.044
growth Strong Asynchrony 1 0 -0.022 0.003 -0.369 0.042
growth Strong Asynchrony 1 0.1 -0.018 0.003 -0.436 0.049
growth Strong Asynchrony 1 0.2 -0.026 0.003 -0.505 0.062
growth Strong Asynchrony 2 0 -0.021 0.003 -0.352 0.038
growth Strong Asynchrony 2 0.1 -0.010 0.001 -0.398 0.044
growth Strong Asynchrony 2 0.2 -0.004 0.001 -0.502 0.055
growth Weak Asynchrony 0 0 -0.029 0.004 -0.259 0.030
growth Weak Asynchrony 0 0.1 -0.017 0.002 -0.192 0.024
growth Weak Asynchrony 0 0.2 -0.021 0.002 -0.210 0.027
growth Weak Asynchrony 1 0 -0.024 0.003 -0.264 0.031
growth Weak Asynchrony 1 0.1 -0.018 0.002 -0.304 0.034
growth Weak Asynchrony 1 0.2 -0.024 0.003 -0.333 0.038
growth Weak Asynchrony 2 0 -0.027 0.003 -0.263 0.030
growth Weak Asynchrony 2 0.1 -0.016 0.002 -0.263 0.031
growth Weak Asynchrony 2 0.2 -0.015 0.002 -0.297 0.035
growth Weak Synchrony 0 0 -0.012 0.002 -0.215 0.026
growth Weak Synchrony 0 0.1 -0.005 0.001 -0.229 0.027
growth Weak Synchrony 0 0.2 -0.004 0.001 -0.248 0.030
growth Weak Synchrony 1 0 -0.009 0.002 -0.218 0.026
growth Weak Synchrony 1 0.1 -0.004 0.001 -0.222 0.026
growth Weak Synchrony 1 0.2 -0.002 0.001 -0.242 0.029
growth Weak Synchrony 2 0 -0.009 0.002 -0.215 0.026
growth Weak Synchrony 2 0.1 -0.003 0.001 -0.183 0.022
growth Weak Synchrony 2 0.2 0.002 0.002 -0.207 0.025
growth Strong Synchrony 0 0 -0.011 0.002 -0.215 0.026
growth Strong Synchrony 0 0.1 -0.003 0.001 -0.263 0.033
growth Strong Synchrony 0 0.2 -0.003 0.001 -0.364 0.045
growth Strong Synchrony 1 0 -0.006 0.001 -0.229 0.029
growth Strong Synchrony 1 0.1 0.004 0.001 -0.282 0.034
growth Strong Synchrony 1 0.2 -0.002 0.001 -0.391 0.046
growth Strong Synchrony 2 0 -0.011 0.002 -0.221 0.025
growth Strong Synchrony 2 0.1 0.004 0.001 -0.230 0.027
growth Strong Synchrony 2 0.2 0.006 0.001 -0.319 0.038
direction interaction Lag Process_error true_within_rlpiCI n perc
decline Strong Asynchrony 0 0 Success 11 1.0000000
decline Strong Asynchrony 0 0.1 Success 11 1.0000000
decline Strong Asynchrony 0 0.2 Success 11 1.0000000
decline Strong Asynchrony 1 0 Success 11 1.0000000
decline Strong Asynchrony 1 0.1 Success 9 0.8181818
decline Strong Asynchrony 1 0.2 Success 8 0.7272727
decline Strong Asynchrony 2 0 Success 11 1.0000000
decline Strong Asynchrony 2 0.1 Success 8 0.7272727
decline Strong Asynchrony 2 0.2 Success 11 1.0000000
decline Weak Asynchrony 0 0 Success 4 0.3636364
decline Weak Asynchrony 0 0.1 Success 11 1.0000000
decline Weak Asynchrony 0 0.2 Success 11 1.0000000
decline Weak Asynchrony 1 0 Success 8 0.7272727
decline Weak Asynchrony 1 0.1 Success 9 0.8181818
decline Weak Asynchrony 1 0.2 Success 6 0.5454545
decline Weak Asynchrony 2 0 Success 5 0.4545455
decline Weak Asynchrony 2 0.1 Success 11 1.0000000
decline Weak Asynchrony 2 0.2 Success 11 1.0000000
decline Weak Synchrony 0 0 Success 7 0.6363636
decline Weak Synchrony 0 0.1 Success 8 0.7272727
decline Weak Synchrony 0 0.2 Success 11 1.0000000
decline Weak Synchrony 1 0 Success 10 0.9090909
decline Weak Synchrony 1 0.1 Success 11 1.0000000
decline Weak Synchrony 1 0.2 Success 11 1.0000000
decline Weak Synchrony 2 0 Success 8 0.7272727
decline Weak Synchrony 2 0.1 Success 11 1.0000000
decline Weak Synchrony 2 0.2 Success 10 0.9090909
decline Strong Synchrony 0 0 Success 8 0.7272727
decline Strong Synchrony 0 0.1 Success 11 1.0000000
decline Strong Synchrony 0 0.2 Success 11 1.0000000
decline Strong Synchrony 1 0 Success 10 0.9090909
decline Strong Synchrony 1 0.1 Success 11 1.0000000
decline Strong Synchrony 1 0.2 Success 4 0.3636364
decline Strong Synchrony 2 0 Success 10 0.9090909
decline Strong Synchrony 2 0.1 Success 11 1.0000000
decline Strong Synchrony 2 0.2 Success 11 1.0000000
stable Strong Asynchrony 0 0 Success 4 0.3636364
stable Strong Asynchrony 0 0.1 Success 11 1.0000000
stable Strong Asynchrony 0 0.2 Success 11 1.0000000
stable Strong Asynchrony 1 0 Success 11 1.0000000
stable Strong Asynchrony 1 0.1 Success 9 0.8181818
stable Strong Asynchrony 1 0.2 Success 7 0.6363636
stable Strong Asynchrony 2 0 Success 11 1.0000000
stable Strong Asynchrony 2 0.1 Success 11 1.0000000
stable Strong Asynchrony 2 0.2 Success 11 1.0000000
stable Weak Asynchrony 0 0 Success 3 0.2727273
stable Weak Asynchrony 0 0.1 Success 11 1.0000000
stable Weak Asynchrony 0 0.2 Success 11 1.0000000
stable Weak Asynchrony 1 0 Success 6 0.5454545
stable Weak Asynchrony 1 0.1 Success 7 0.6363636
stable Weak Asynchrony 1 0.2 Success 6 0.5454545
stable Weak Asynchrony 2 0 Success 3 0.2727273
stable Weak Asynchrony 2 0.1 Success 11 1.0000000
stable Weak Asynchrony 2 0.2 Success 11 1.0000000
stable Weak Synchrony 0 0 Success 6 0.5454545
stable Weak Synchrony 0 0.1 Success 7 0.6363636
stable Weak Synchrony 0 0.2 Success 11 1.0000000
stable Weak Synchrony 1 0 Success 6 0.5454545
stable Weak Synchrony 1 0.1 Success 7 0.6363636
stable Weak Synchrony 1 0.2 Success 11 1.0000000
stable Weak Synchrony 2 0 Success 6 0.5454545
stable Weak Synchrony 2 0.1 Success 10 0.9090909
stable Weak Synchrony 2 0.2 Success 11 1.0000000
stable Strong Synchrony 0 0 Success 6 0.5454545
stable Strong Synchrony 0 0.1 Success 8 0.7272727
stable Strong Synchrony 0 0.2 Success 11 1.0000000
stable Strong Synchrony 1 0 Success 8 0.7272727
stable Strong Synchrony 1 0.1 Success 11 1.0000000
stable Strong Synchrony 1 0.2 Success 2 0.1818182
stable Strong Synchrony 2 0 Success 6 0.5454545
stable Strong Synchrony 2 0.1 Success 11 1.0000000
stable Strong Synchrony 2 0.2 Success 11 1.0000000
growth Strong Asynchrony 0 0 Success 4 0.3636364
growth Strong Asynchrony 0 0.1 Success 11 1.0000000
growth Strong Asynchrony 0 0.2 Success 11 1.0000000
growth Strong Asynchrony 1 0 Success 4 0.3636364
growth Strong Asynchrony 1 0.1 Success 6 0.5454545
growth Strong Asynchrony 1 0.2 Success 5 0.4545455
growth Strong Asynchrony 2 0 Success 8 0.7272727
growth Strong Asynchrony 2 0.1 Success 11 1.0000000
growth Strong Asynchrony 2 0.2 Success 11 1.0000000
growth Weak Asynchrony 0 0 Success 3 0.2727273
growth Weak Asynchrony 0 0.1 Success 9 0.8181818
growth Weak Asynchrony 0 0.2 Success 7 0.6363636
growth Weak Asynchrony 1 0 Success 4 0.3636364
growth Weak Asynchrony 1 0.1 Success 5 0.4545455
growth Weak Asynchrony 1 0.2 Success 3 0.2727273
growth Weak Asynchrony 2 0 Success 4 0.3636364
growth Weak Asynchrony 2 0.1 Success 8 0.7272727
growth Weak Asynchrony 2 0.2 Success 9 0.8181818
growth Weak Synchrony 0 0 Success 8 0.7272727
growth Weak Synchrony 0 0.1 Success 10 0.9090909
growth Weak Synchrony 0 0.2 Success 11 1.0000000
growth Weak Synchrony 1 0 Success 11 1.0000000
growth Weak Synchrony 1 0.1 Success 11 1.0000000
growth Weak Synchrony 1 0.2 Success 11 1.0000000
growth Weak Synchrony 2 0 Success 10 0.9090909
growth Weak Synchrony 2 0.1 Success 10 0.9090909
growth Weak Synchrony 2 0.2 Success 11 1.0000000
growth Strong Synchrony 0 0 Success 8 0.7272727
growth Strong Synchrony 0 0.1 Success 11 1.0000000
growth Strong Synchrony 0 0.2 Success 11 1.0000000
growth Strong Synchrony 1 0 Success 11 1.0000000
growth Strong Synchrony 1 0.1 Success 9 0.8181818
growth Strong Synchrony 1 0.2 Success 11 1.0000000
growth Strong Synchrony 2 0 Success 8 0.7272727
growth Strong Synchrony 2 0.1 Success 11 1.0000000
growth Strong Synchrony 2 0.2 Success 11 1.0000000

Do the confidence intervals widen when there is more noise introduced into the time series?

Expectation: in each case, the width of the CI should increase with process error. The width should also increase when there is strong asynchrony or synchrony.

Not really…